A239832 Number of partitions of n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.

0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 7, 6, 11, 11, 17, 19, 27, 31, 41, 51, 62, 79, 95, 121, 142, 182, 212, 269, 314, 393, 459, 570, 665, 816, 958, 1160, 1364, 1639, 1928, 2297, 2706, 3200, 3768, 4434, 5212, 6105, 7170, 8361, 9799, 11396, 13322, 15450, 18022

Offset: 0

Author

Clark Kimberling, Mar 29 2014

Keywords

• nonn
• easy

Comments

Let c(n) be the number of partitions of n having 1 more odd part than even, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are odd. Then c(n) = a(n+1) for n >= 0.

Examples

			The three partitions counted by a(10) are [10], [4,1,2,1,2], and [2,3,2,1,2].
		

Crossrefs

Cf. A239833, A239835, A045931, A239871.

Column k=-1 of A240009.

Programs

Mathematica

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions *)
Table[Length[p[n]], {n, 0, 30}]  (* A239832 *)
(* Peter J. C. Moses, Mar 10 2014 *)